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">[1]</span></a></sup>只有四种，DFT不是真正的FT”，我当时无法理解：“从数学上看，傅里叶变换不就是时域和频域的一种映射关系吗，这种映射关系只是数学上的一种形式，那DFT为何不是呢？”</p>
<p>接触了数字滤波器，对此才略微理解。系统的单位冲激响应和系统函数同样能表示这个系统，但若对h(n)进行DFT，得到的并不是表示这个系统的系统函数H(z)，而是H(z)的抽样。</p>
<p>也就是说，傅里叶变换应该满足<strong>一一映射</strong>的关系，h(n)便对应H(z)，看见时域便见频域，<strong>时域和频域就好似光与影，是同一事物的两面，不可分割</strong><sup id="fnref:2"><a href="#fn:2" rel="footnote"><span class="hint--top hint--error hint--medium hint--rounded hint--bounce" aria-label="“光与影”是一种浪漫的说法，这个比喻并不恰当。
">[2]</span></a></sup>。而DFT根据变换长度的不同，h(n)会对应多个H(k)，这便是DFT隐含周期性的含义，若考虑时域和频域的周期性，那么DFT便可满足一一映射的关系了，而此时，DFT揭示的是周期的时域和周期的频域之间一一映射的关系。</p>
<p>同时，H(k)可以还原出h(n)，h(n)又和H(z)一一对应，因此只要通过H(z)的取样，便能还原出H(z)<sup id="fnref:3"><a href="#fn:3" rel="footnote"><span class="hint--top hint--error hint--medium hint--rounded hint--bounce" aria-label="内插公式
">[3]</span></a></sup>，这个性质揭示了系统的某种对称性<sup id="fnref:4"><a href="#fn:4" rel="footnote"><span class="hint--top hint--error hint--medium hint--rounded hint--bounce" aria-label="FFT也是基于此">[4]</span></a></sup>。</p>
<div id="footnotes"><hr><div id="footnotelist"><ol style="list-style: none; padding-left: 0; margin-left: 40px"><li id="fn:1"><span style="display: inline-block; vertical-align: top; padding-right: 10px; margin-left: -40px">1.</span><span style="display: inline-block; vertical-align: top; margin-left: 10px;">这里的FT指的是广义的傅里叶变换，而非专指时域连续信号到频域的变换。<a href="#fnref:1" rev="footnote"> ↩</a></span></li><li id="fn:2"><span style="display: inline-block; vertical-align: top; padding-right: 10px; margin-left: -40px">2.</span><span style="display: inline-block; vertical-align: top; margin-left: 10px;">“光与影”是一种浪漫的说法，这个比喻并不恰当。<a href="#fnref:2" rev="footnote"> ↩</a></span></li><li id="fn:3"><span style="display: inline-block; vertical-align: top; padding-right: 10px; margin-left: -40px">3.</span><span style="display: inline-block; vertical-align: top; margin-left: 10px;">内插公式<a href="#fnref:3" rev="footnote"> ↩</a></span></li><li id="fn:4"><span style="display: inline-block; vertical-align: top; padding-right: 10px; margin-left: -40px">4.</span><span style="display: inline-block; vertical-align: top; margin-left: 10px;">FFT也是基于此<a href="#fnref:4" rev="footnote"> ↩</a></span></li></ol></div></div></article><div class="tag_share"><div class="post-meta__tag-list"><a class="post-meta__tags" href="/tags/%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2/">傅里叶变换</a></div><div class="post_share"><div class="social-share" data-image="/linear-gradient(20deg,#334d50,#cbcaa5)" data-sites="facebook,twitter,wechat,weibo,qq"></div><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/butterfly-extsrc/sharejs/dist/css/share.min.css" media="print" onload="this.media='all'"><script src="https://cdn.jsdelivr.net/npm/butterfly-extsrc/sharejs/dist/js/social-share.min.js" defer></script></div></div><div class="post-reward"><div class="reward-button"><i class="fas fa-qrcode"></i> Donate</div><div class="reward-main"><ul class="reward-all"><li class="reward-item"><a href="/img/Wechat.jpg" 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